Two%20Dimensional%20Gauge%20Theories%20and%20Quantum%20Integrable%20Systems

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Two Dimensional Gauge Theoriesand Quantum
Integrable Systems

• Nikita Nekrasov
• IHES
• Imperial College
• April 10, 2008

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Based on

• NN, S.Shatashvili, to appear
• Prior work
• E.Witten, 1992
• A.Gorsky, NN J.Minahan, A.Polychronakos
• M.Douglas 1993-1994 A.Gerasimov 1993
• G.Moore, NN, S.Shatashvili 1997-1998
• A.Losev, NN, S.Shatashvili 1997-1998
• A.Gerasimov, S.Shatashvili 2006-2007

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We are going to relate 2,3, and 4 dimensional
susy gauge theorieswith four supersymmetries
N1 d4

• And quantum integrable systems
• soluble by Bethe Ansatz techniques.

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Mathematically speaking, the cohomology,
K-theory and elliptic cohomology of various gauge
theory moduli spaces, like moduli of flat
connections and instantons

• And quantum integrable systems
• soluble by Bethe Ansatz techniques.

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• For example, we shall relate the
• XXX Heisenberg magnet
• and
• 2d N2 SYM theory
• with some matter

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(pre-)History

• In 1992 E.Witten studied two dimensional
  Yang-Mills theory with the goal to understand the
  relation between the physical and topological
  gravities in 2d.

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(pre-)History

• There are two interesting kinds of
• Two dimensional Yang-Mills theories

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Yang-Mills theories in 2d

• (1)
• Cohomological YM
• twisted N2 super-Yang-Mills theory,
• with gauge group G,
• whose BPS (or TFT) sector is related to
• the intersection theory on
• the moduli space MG of
• flat G-connections on
• a Riemann surface

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Yang-Mills theories in 2d

• N2 super-Yang-Mills theory

Field content
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Yang-Mills theories in 2d

• (2)
• Physical YM
• N0 Yang-Mills theory, with gauge group G
• The moduli space MG of flat G-connections
• minima of the action
• The theory is exactly soluble (A.Migdal) with the
  help of the Polyakov lattice YM action

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Yang-Mills theories in 2d

• Physical YM

Field content
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Yang-Mills theories in 2d

• Witten found a way to map the BPS sector of the
  N2 theory to the N0 theory.
• The result is

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Yang-Mills theories in 2d

• Two dimensional Yang-Mills partition function is
  given by the explicit sum

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Yang-Mills theories in 2d

• In the limit
• the partition function computes the volume of MG

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Yang-Mills theories in 2d

• Wittens approach add twisted superpotential and
  its conjugate

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Yang-Mills theories in 2d

• Take a limit

In the limit the fields are infinitely massive
and can be integrated out one is left with the
field content of the physical YM theory
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Yang-Mills theories in 2d

• Both physical and cohomological Yang-Mills
• theories define topological field theories (TFT)

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Yang-Mills theories in 2d

• Both physical and cohomological Yang-Mills
• theories define topological field theories (TFT)

Vacuum states deformations quantum mechanics
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YM in 2d and particles on a circle
Physical YM is explicitly equivalent to a
quantum mechanical model free fermions on a
circle
Can be checked by a partition function on a
two-torus
Gross Douglas
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YM in 2d and particles on a circle
Physical YM is explicitly equivalent to a
quantum mechanical model free fermions on a
circle
States are labelled by the partitions, for GU(N)
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YM in 2d and particles on a circle
For N2 YM these free fermions on a circle
Label the vacua of the theory deformed by twisted
superpotential W
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YM in 2d and particles on a circle
The fermions can be made interacting by adding a
localized matter for example a time-like Wilson
loop in some representation V of the gauge group
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YM in 2d and particles on a circle
One gets Calogero-Sutherland (spin) particles on
a circle (1993-94) A.Gorsky,NN
J.Minahan,A.Polychronakos
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History

• In 1997 G.Moore, NN and S.Shatashvili studied
  integrals over
• various hyperkahler quotients,
• with the aim to understand
• instanton integrals in
• four dimensional gauge theories

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History

• In 1997 G.Moore, NN and S.Shatashvili studied
  integrals over
• various hyperkahler quotients,
• with the aim to understand
• instanton integrals in
• four dimensional gauge theories
• This eventually led to the derivation in 2002 of
• the Seiberg-Witten solution of N2 d4 theory

Inspired by the work of H.Nakajima
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Yang-Mills-Higgs theory

• Among various examples, MNS studied Hitchins
  moduli space MH

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Yang-Mills-Higgs theory

• Unlike the case of two-dimensional
• Yang-Mills theory where the moduli space MG is
  compact,
• Hitchins moduli space is non-compact
• (it is roughly TMG modulo subtleties) and the
  volume is infinite.

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Yang-Mills-Higgs theory

• In order to cure this infnity in a reasonable way
  MNS used the U(1) symmetry of MH

The volume becomes a DH-type expression
Where H is the Hamiltonian
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Yang-Mills-Higgs theory

• Using the supersymmetry and localization
• the regularized volume of MH
• was computed with the result

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Yang-Mills-Higgs theory

• Where the eigenvalues solve the equations

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YMH and NLS

• The experts would immediately recognise the
• Bethe ansatz (BA) equations for
• the non-linear Schroedinger theory (NLS)

NLS large spin limit of the SU(2) XXX spin chain
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YMH and NLS

• Moreover the NLS Hamiltonians
• are the 0-observables of the theory, like

The VEV of the observable The eigenvalue of
the Hamiltonian
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YMH and NLS

• Since 1997 nothing came out of
• this result.
• It could have been simply a coincidence.
• .

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In 2006 A.Gerasimov and S.Shatashvili have
revived the subject

• History

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YMH and interacting particles

• GS noticed that YMH theory viewed as TFT is
  equivalent to the quantum Yang system
• N particles on a circle with delta-interaction

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YMH and interacting particles

• Thus YM with the matter -- fermions with
  pair-wise interaction

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History

• More importantly,
• GS suggested that TFT/QIS equivalence is much
  more universal

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Today

• We shall rederive the result of MNS from a modern
  perspective
• Generalize to cover virtually all BA soluble
  systems both with finite and infinite spin
• Suggest natural extensions of the BA equations

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Hitchin equations

• Solutions can be viewed as the susy field
  configurations for
• the N2 gauged linear sigma model

For adjoint-valued linear fields
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Hitchin equations

• The moduli space MH of solutions is a hyperkahler
  manifold
• The integrals over MH are computed by the
  correlation functions of
• an N2 d2 susy gauge theory

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Hitchin equations

• The kahler form on MH comes from
• twisted tree level superpotential
• The epsilon-term comes
  from
• a twisted mass of the matter multiplet

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Generalization

• Take an N2 d2 gauge theory with matter,
• In some representation R
• of the gauge group G

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Generalization

• Integrate out the matter fields,
• compute the effective (twisted)
• super-potential
• on the Coulomb branch

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Mathematically speaking

• Consider the moduli space MR of R-Higgs pairs
• with gauge group G

Up to the action of the complexified gauge group
GC
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Mathematically speaking

• Stability conditions

Up to the action of the compact gauge group G
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Mathematically speaking

• Pushforward the unit class down to
• the moduli space MG of GC-bundles
• Equivariantly with respect to the action
• of the global symmetry group K on MR

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Mathematically speaking

• The pushforward can be expressed in terms of the
  Donaldson-like classes of
• the moduli space MG
• 2-observables and 0-observables

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Mathematically speaking

• The pushforward can be expressed in terms of the
  Donaldson-like classes of
• the moduli space MG
• 2-observables and 0-observables

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Mathematically speaking

• The masses are the equivariant parameters
• For the global symmetry group K

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Vacua of the gauge theory
For G U(N)

• Due to quantization of the gauge flux

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Vacua of the gauge theory
For G U(N)

• Equations familiar from yesterdays lecture

partitions
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Vacua of the gauge theory

• Familiar example CPN model

(N1) chiral multiplet of charge 1 Qi i1, ,
N1 U(1) gauge group
Field content
Effective superpotential
N1 vacuum
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Vacua of gauge theory
Another example

• Gauge group
• GU(N)
• Matter chiral multiplets
• 1 adjoint, mass
• fundamentals, mass
• anti-fundamentals, mass

Field content
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Vacua of gauge theory
Effective superpotential
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Vacua of gauge theory
Equations for vacua
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Vacua of gauge theory
Non-anomalous case
Redefine
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Vacua of gauge theory
Vacua
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Gauge theory -- spin chain
Identical to the Bethe ansatz equations for spin
XXX magnet
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Gauge theory -- spin chain
Vacua eigenstates of the Hamiltonian
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Table of dualities

• XXX spin chain
• SU(2)
• L spins
• N excitations

U(N) d2 N2 Chiral multiplets 1 adjoint L
fundamentals L anti-fund.
Special masses!
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Table of dualities mathematically speaking

• XXX spin chain
• SU(2)
• L spins
• N excitations

(Equivariant) Intersection theory on MR for
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Table of dualities

• XXZ spin chain
• SU(2)
• L spins
• N excitations

U(N) d3 N1 Compactified on a circle Chiral
multiplets 1 adjoint L fundamentals L anti-fund.
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Table of dualities mathematically speaking

• XXZ spin chain
• SU(2)
• L spins
• N excitations

Equivariant K-theory of the moduli space MR
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Table of dualities

• XYZ spin chain
• SU(2), L 2N spins
• N excitations

U(N) d4 N1 Compactified on a 2-torus
elliptic curve E Chiral multiplets 1
adjoint L 2N fundamentals L 2N anti-fund.
Masses wilson loops of the
flavour group points on the Jacobian of E
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Table of dualities mathematically speaking

• XYZ spin chain
• SU(2), L 2N spins
• N excitations

Elliptic genus of the moduli space MR
Masses K bundle over E points on the BunK of E
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Table of dualities

• It is remarkable that the spin chain has
• precisely those generalizations
• rational (XXX), trigonometric (XXZ) and elliptic
  (XYZ)
• that can be matched to the 2, 3, and 4 dim cases.
  

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Algebraic Bethe Ansatz
Faddeev et al.

• The spin chain is solved algebraically using
  certain operators,
• Which obey exchange commutation relations

Faddeev-Zamolodchikov algebra
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Algebraic Bethe Ansatz

• The eigenvectors, Bethe vectors, are obtained by
  applying these operators to the  fake  vacuum.

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ABA vs GAUGE THEORY

• For the spin chain it is natural to fix L total
  number of spins
• and consider various N excitation levels
• In the gauge theory context N is fixed.

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ABA vs GAUGE THEORY

• However, if the theory is embedded into string
  theory via brane realization
• then changing N is easy
• bring in an extra brane.

Hanany-Hori02
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ABA vs GAUGE THEORY

• Mathematically speaking
• We claim that the Algebraic Bethe Ansatz is most
  naturally related to the derived category of the
  category of coherent sheaves on some local CY

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ABA vs STRING THEORY

• THUS
• B is for BRANE!

is for location!
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More general spin chains

• The SU(2) spin chain
• has generalizations to
• other groups and representations.
• I quote the corresponding
• Bethe ansatz equations
• from N.Reshetikhin

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General groups/reps

• For simply-laced group H of rank r

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General groups/reps

• For simply-laced group H of rank r

Label representations of the Yangian of H
A.N.Kirillov-N.Reshetikhin modules
Cartan matrix of H
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General groups/repsfrom GAUGE THEORY

• Take the Dynkin diagram corresponding to H
• A simply-laced group of rank r

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QUIVER GAUGE THEORY

• Symmetries

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QUIVER GAUGE THEORY

• Symmetries

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QUIVER GAUGE THEORYCharged matter
Adjoint chiral multiplet
Fundamental chiral multiplet
Anti-fundamental chiral multiplet
Bi-fundamental chiral multiplet
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QUIVER GAUGE THEORY

• Matter fields adjoints

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QUIVER GAUGE THEORY

• Matter fields
• fundamentalsanti-fundamentals

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QUIVER GAUGE THEORY

• Matter fields bi-fundamentals

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QUIVER GAUGE THEORY

• Quiver gauge theory full content

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QUIVER GAUGE THEORY MASSES

• Adjoints

i
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QUIVER GAUGE THEORY MASSES

• Fundamentals
• Anti-fundamentals

i
a 1, . , Li
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QUIVER GAUGE THEORY MASSES

• Bi-fundamentals

j
i
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QUIVER GAUGE THEORY

• What is so special about these masses?

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QUIVER GAUGE THEORY

• From the gauge theory point of view nothing
  special..

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QUIVER GAUGE THEORY

• The mass puzzle!

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The mass puzzle

• The Bethe ansatz -- like equations

Can be written for an arbitrary matrix
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The mass puzzle

• However the Yangian symmetry Y(H) would get
  replaced by some ugly infinite-dimensional
   free  algreba without nice representations

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The mass puzzle

• Therefore we conclude that our choice of masses
  is dictated by the hidden symmetry -- that of the
  dual spin chain

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The Standard Model has many free parameters

• Among them are the fermion masses
• Is there a (hidden) symmetry principle behind
  them?

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The Standard Model has many free parameters

• In the supersymmetric models
• we considered
• the mass tuning
• can be  explained 
• using a duality to some
• quantum integrable system

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Further generalizationsSuperpotential from
prepotential
Tree level part
Flux superpotential (Losev,NN, Shatashvili97)
Induced by twist
The N2 theory on R2 X S2
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Superpotential from prepotential
Magnetic flux
Electric flux
In the limit of vanishing S2 the magnetic flux
should vanish
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Instanton corrected BA equations
Effective S-matrix contains 2-body, 3-body,
interactions
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Instanton corrected BA equations
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Instanton corrected BA equations
The prepotential of the low-energy effective
theory Is governed by a classical (holomorphic)
integrable system
Donagi-Witten95
Liouville tori Jacobians of Seiberg-Witten
curves
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Classical integrable systemvsQuantum integrable
system
That system is quantized when the gauge theory is
subject to the Omega-background
NN02 NN,Okounkov03 Braverman03
Our quantum system is different!
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Blowing up the two-sphere

• Wall-crossing phenomena
• (new states, new solutions)

Something for the future
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Naturalness of our quivers

• Somewhat unusual matter content
• Branes at orbifolds typically lead to smth like

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Naturalness of our quivers

• This picture would arise in the
• sa(i) ? 0
• limit

BA for QCD Faddeev-Korchemsky94
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Naturalness of our quivers

• Other quivers?

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Naturalness of our quivers

• Possibly with the help of K.Saitos construction
  

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CONCLUSIONS

1. We found the Bethe Ansatz equations are the
   equations describing the vacuum configurations of
   certain quiver gauge theories in two dimensions
2. The duality to the spin chain requires certain
   relations between the masses of the matter fields
   to be obeyed. This could have phenomenological
   consequences.

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CONCLUSIONS

• 3. The algebraic Bethe ansatz seems to provide a
  realization of the brane creation operators --
  something of major importance both for
  topological and physical string theories
• 4. Obviously this is a beginning of a beautiful
  story.